What is the integral of (x^2)/(3x-1) ?

The integral of (x^2)/(3x-1) is 1/54 (9x^2+6x+2ln|3x-1|-3)+C

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integral of (x^2)/(3x-1)

\int \frac{x ^{2}}{3 x - 1} d x

\frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) + C

Solution steps

\int \frac{x ^{2}}{3 x - 1} d x

Apply u-substitution

: \int \frac{( u + 1 ) ^{2}}{27 u} d u

= \int \frac{( u + 1 ) ^{2}}{27 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{27} \cdot \int \frac{( u + 1 ) ^{2}}{u} d u

Expand

\frac{( u + 1 ) ^{2}}{u} : u + 2 + \frac{1}{u}

= \frac{1}{27} \cdot \int u + 2 + \frac{1}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{27} (\int u d u + \int 2 d u + \int \frac{1}{u} d u)

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{27} (\frac{u ^{2}}{2} + 2 u + ln ∣ u ∣)

Substitute back

u = 3 x - 1

= \frac{1}{27} (\frac{( 3 x - 1 ) ^{2}}{2} + 2 (3 x - 1) + ln ∣ 3 x - 1 ∣)

Simplify

\frac{1}{27} (\frac{( 3 x - 1 ) ^{2}}{2} + 2 (3 x - 1) + ln ∣ 3 x - 1 ∣) : \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3)

= \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3)

Add a constant to the solution

= \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) + C

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What is the integral of (x^2)/(3x-1) ?
The integral of (x^2)/(3x-1) is 1/54 (9x^2+6x+2ln|3x-1|-3)+C